Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets

TL;DR

This paper introduces a new uncertainty relation for pairs of general signal sets, improving sparsity thresholds for signal recovery and providing probabilistic guarantees for dictionary-based sparse signal reconstruction.

Contribution

It develops an uncertainty relation for two general signal sets, leading to enhanced sparsity thresholds and probabilistic recovery guarantees beyond existing methods.

Findings

Improved sparsity thresholds up to twice the previous $(1+1/d)/2$-threshold.

Probabilistic guarantees for dictionary pairs in sparse recovery.

Insights into which dictionary parts to randomize for better recovery.

Abstract

We present an uncertainty relation for the representation of signals in two different general (possibly redundant or incomplete) signal sets. This uncertainty relation is relevant for the analysis of signals containing two distinct features each of which can be described sparsely in a suitable general signal set. Furthermore, the new uncertainty relation is shown to lead to improved sparsity thresholds for recovery of signals that are sparse in general dictionaries. Specifically, our results improve on the well-known $(1+1/d)/2$-threshold for dictionaries with coherence $d$ by up to a factor of two. Furthermore, we provide probabilistic recovery guarantees for pairs of general dictionaries that also allow us to understand which parts of a general dictionary one needs to randomize over to "weed out" the sparsity patterns that prohibit breaking the square-root bottleneck.